Electrostatic Damping Research Articles

Enhancing the Shock Response Performance of Micromachined Silicon Resonant Accelerometers by Electrostatic Active Damping Control.

This paper presents a micromachined silicon resonant accelerometer based on electrostatic active damping control, which can improve the shock response performance of the accelerometer. In the accelerometer, an electrostatic active damping structure and damping control circuit are designed to improve the equivalent damping coefficient of the system. System-level Simulink modeling and simulation of the accelerometer with an electrostatic active damping closed-loop control link were carried out. The simulation results indicate that the system can quickly return to normal output without an obvious vibration process after the shock. The fabricated and packaged accelerometer was connected to an external test circuit for shock performance testing. The stabilization time of the accelerometer after a 100 g, 3–5 ms half-sine shock was reduced from 19.8 to 5.6 s through use of the damping control. Furthermore, the change in deviation before and after the shock without damping control was 0.8197 mg, whereas it was 0.1715 mg with damping control. The experimental results demonstrate that the electrostatic active damping control can effectively improve the dynamic performance of the micromachined silicon resonant accelerometer.

Electrostatic frequency reduction: A negative stiffness mechanism for measuring dissipation in a mechanical oscillator at low frequency.

Broadband seismometers and gravitational wave detectors make use of mechanical resonators with a high quality factor to reduce Brownian noise. At low frequency, Brownian noise is ultimately dominated by internal friction in the suspension, which has a 1/f noise compared with the white noise arising from viscous dissipation. Internal friction is typically modeled as a frequency-dependent loss and can be challenging to measure reliably through experiment. In this work, we present the physics and experimental implementation of electrostatic frequency reduction (EFR) in a mechanical oscillator-a method to measure dissipation as a function of frequency. By applying a high voltage to two parallel capacitor plates, with the center plate being a suspended mass, an electrostatic force is created that acts as a negative stiffness mechanism to reduce the system's resonance frequency. Through EFR, the loss angle can be measured as a function of frequency by measuring amplitude decay response curves for a range of applied voltages. We present experimental measurements of the loss angle for three metal helical extension springs in the nominal frequency range 0.7-2.9 Hz at 0.2 Hz intervals, demonstrating the possibility for fine adjustment of the resonance frequency for loss angle measurements. A quality factor proportional to the resonance frequency squared was measured, an indication that internal friction and other non-viscous dissipation elements, such as electrostatic damping, were the prominent loss mechanisms in our experiments. Finally, we consider the implications of Brownian noise arising from internal friction on a low 1/f noise seismometer.

A predictive model for electrostatic energy harvesters with impact-based frequency up-conversion

This paper reports a predictive model for vibration-to-electrical energy harvesters based on an in-plane, gap-closing variable capacitor with frequency up-conversion triggered by the impact between the electrodes. Since the output power is proportional to the output frequency, rectifying low-frequency ambient vibrations (1–50 Hz) to high-frequency electrical signals (200–600 Hz) increases the power output. While such a device has been previously reported experimentally, this is the first time a model able to predict the experimental data has been described. The model is based on lumped approximation. The central area supporting the mobile electrodes, or the shuttle mass, is represented by a point mass suspended by springs and has its own equation of motion. The motion of the electrodes attached to the shuttle is described by a set of two equations, each associated with a distinct dynamic mode. In these equations, the electrodes are represented by an equivalent mass and spring constant. The first equation describes the separate motion of mobile and fixed electrodes. In this mode, the electrodes experience damped free vibration due to the electrostatic and air damping between them. The second equation describes the combined motion when the two set of electrodes move together. In this mode the air damping forces between the collided electrodes is eliminated and the electrostatic force is kept constant. The motion equations are solved simultaneously with Kirchhoff’s law to compute the voltage drop across a resistor in series with the variable capacitor and a DC bias voltage source. Predictions are shown to be consistent with the experimental results, and frequency up-conversion effects are observed with exponentially decaying voltage amplitude as seen in experiments. A parametric study is also carried out to identify main parameters that affect the up-conversion, laying the foundation for future design optimization to maximize the power output.

Demonstration of dynamic thermal compensation for parametric instability suppression in Advanced LIGO

Advanced LIGO and other ground-based interferometric gravitational-wave detectors use high laser power to minimize shot noise and suspended optics to reduce seismic noise coupling. This can result in an opto-mechanical coupling which can become unstable and saturate the interferometer control systems. The severity of these parametric instabilities scales with circulating laser power and first hindered LIGO operations in 2014. Static thermal tuning and active electrostatic damping have previously been used to control parametric instabilities at lower powers but are insufficient as power is increased. Here we report the first demonstration of dynamic thermal compensation to avoid parametric instability in an Advanced LIGO detector. Annular ring heaters that compensate central heating are used to tune the optical mode away from multiple problematic mirror resonance frequencies. We develop a single-cavity approximation model to simulate the optical beat note frequency during the central heating and ring heating transient. An experiment of dynamic ring heater tuning at the LIGO Livingston detector was carried out at 170 kW circulating power and, in agreement with our model, the third order optical beat note is controlled to avoid instability of the 15 and 15.5 kHz mechanical modes. We project that dynamic thermal compensation with ring heater input conditioning can be used in parallel with acoustic mode dampers to control the optical mode transient and avoid parametric instability of these modes up to Advanced LIGO’s design circulating power of 750 kW. The experiment also demonstrates the use of three mode interaction monitoring as a sensor of the cavity geometry, used to maintain the g-factor product to g 1 g 2 = 0.829 ± 0.004.

Quantum algorithm for the Vlasov equation

The Vlasov-Maxwell system of equations, which describes classical plasma physics, is extremely challenging to solve, even by numerical simulation on powerful computers. By linearizing and assuming a Maxwellian background distribution function, we convert the Vlasov-Maxwell system into a Hamiltonian simulation problem. Then for the limiting case of electrostatic Landau damping, we design and verify a quantum algorithm, appropriate for a future error-corrected universal quantum computer. While the classical simulation has costs that scale as $\mathcal{O}(N_v t)$ for a velocity grid with $N_v$ grid points and simulation time $t$, our quantum algorithm scales as $\mathcal{O}(\text{polylog}(N_v) t/\delta)$ where $\delta$ is the measurement error, and weaker scalings have been dropped. Extensions, including electromagnetics and higher dimensions, are discussed. A quantum computer could efficiently handle a high-resolution, six-dimensional phase-space grid, but the $1/\delta$ cost factor to extract an accurate result remains a difficulty. This paper provides insight into the possibility of someday achieving efficient plasma simulation on a quantum computer.

Wavelet based damage identification and dynamic pull-in instability analysis of electrostatically actuated coupled domain microsystems using generalized differential quadrature method

Abstract In this study, the effects of edged-crack on dynamic pull-in instability of fixed-fixed microbeams under suddenly applied electrostatic excitations and nonlinear squeeze film damping are investigated. The deformable electrode is modeled based on size dependent Euler–bernoulli beam formulations with modified couple stress theory. Also fluid interaction with electrode is modeled by nonlinear Reynolds equation. The model accounts for the von Karman nonlinear strains and edge crack in the domain. The edged crack is modeled as a massless torsional spring and is incorporated in the continuity equations of micro-beam formulations. Employing the Hamilton's principle, the governing nonlinear equations are attained and are then discretized by generalized differential quadrature method. This approach is combined with Newmark time integration and direct iteration method and is employed to investigate dynamic pull-in instability of the microsystem. The stability analysis of the cracked microbeam is also implemented by evaluating the largest Lyapunov exponent, the sign of which describes the character of the time dependent response. Wavelet transform based on Gabor function is applied for identifying the damage in microstructure. The location of damage is determined from the sudden peaks in the spatial variation of the transformed response of damped microbeam by this algorithm. Finally, a comprehensive study is carried out to examine the influence of the crack depth and crack position on pull-in instability characteristics of microbeams. Also, the efficiency of proposed wavelet analysis for damage detection in microstructures is assessed.

Triboelectric energy harvesting from the vibro-impact of three cantilevered beams

Abstract A new system that harvests vibration energy through triboelectric and electrostatic effects from the vibro-impact of three parallel cantilevered beams is theoretically studied and experimentally validated. Two modelling methods of the mechanical vibration, which differ in the consideration of the eccentricity of the tip mass, are presented and compared with the experimental results for validation. An experimentally identified and numerically fitted relationship between the just-before-impact velocity and the coefficient of restitution is used to model the impact. The electrical model of the triboelectric energy harvester is then established. The ordinary differential equation (ODE) modelling the electrical output of the triboelectric energy harvester is found to be stiff. Two schemes, which are based on the TR (the trapezoidal rule) and TR-BDF2 (the composite of the trapezoidal rule and the second order backward differentiation formula) methods, are proposed to solve the non-smooth mechanical system and the stiff electrical system. The experimental investigations of the effects of electrostatic force and air damping between cantilever tips suggest that their effects are negligible. This enables the mechanical and electrical systems to be modelled as uncoupled. Good agreement between numerical and experimental results is found, especially for vibration responses. The modelling method, not involving the eccentricity of the tip mass, offers better agreement with experiments. A new numerical scheme combining the TR and TR-BDF2 methods is found to be superior because it is the only method that can solve the electrical system and produce stable results around the impacts. Larger amplitude vibration associated with smaller tip mass ratios does not guarantee higher electricity generation while larger mass ratios perform better at higher frequencies and have a wider effective bandwidth over which their electricity generation performance is superior. Both the responses of vibration and electrical output increases linearly with the excitation level while the relationship between the tribo-charge surface density and the excitation amplitude is found to be quadratic.

Flux-balance Vlasov simulation with filamentation filtration

An Eulerian phase-space discretized Vlasov–Poisson simulation code that incorporates filamentation filtration is introduced. The code is based on the flux-balance method, which is extended to include the filtration. A novel method for removing a localized numerical instability at the velocity boundaries is introduced and it is shown that the code is otherwise stable for a limited but effective range of filamentation filter widths. It is demonstrated that filamentation filtration does not remove the recurrence problem; recurrence does arise in the absence of filamentation. Simulations of one-dimensional, electrostatic Landau damping, and two-stream instability are presented. For these simulations, favorable comparisons with the results of a well-established Fourier–Fourier transformed spectral code are discussed.

Sensing and cooling of a nanomechanical resonator with an electron beam stimulated internal feedback and a capacitive force

A model for the cooling properties of a nanocantilever by a free electron beam is presented for a capacitive interaction. The optimal parameters for position sensing and cooling applications are estimated from previous experimental conditions. In particular, we demonstrate that a purely capacitive force and an electron beam stimulated internal feedback can lower the temperature of a nanocantilever by several orders of magnitude, in striking contrast with the conventional electrostatic damping regime. We propose a step by step protocol to extract the interdependent parameters of the experiments. This work will aid future developments of ultra-sensitive force sensors in electron microscopes.

New Stability Method of a Multirate Controller for a Three-Axis High-<inline-formula> <tex-math notation="LaTeX">$Q$ </tex-math> </inline-formula> MEMS Accelerometer With Simultaneous Electrostatic Damping

Over the past years, cutting-edge advances in electronics and microfabrication have allowed the integration of multiple sensors within integrated analog and digital circuits to design microelectromechanical systems. The multiple sensor integration or sensor fusion enables both cost and surface reduction, while maintaining high performances. This paper presents a new control system for an underdamped three-axis accelerometer, which allows the co-integration in the same cavity with a three-axis Coriolis gyroscope to design a six degrees-of-freedom combo sensor. The accelerometer analog front end consumes $300~\mu \text{A}$ from a 1.6V power supply and is able to reach its steady state in $800~\mu \text{s}$ compared with a 400ms open loop and no damping configuration. The transducer control is implemented using a simultaneous multirate electrostatic damping method. To conclude on the closed-loop system stability, an innovative approach based on the multirate signal processing theory has been developed.

Multidimensional Vlasov–Poisson Simulations with High-order Monotonicity- and Positivity-preserving Schemes

Abstract We develop new numerical schemes for Vlasov–Poisson equations with high-order accuracy. Our methods are based on a spatially monotonicity-preserving (MP) scheme and are modified suitably so that the positivity of the distribution function is also preserved. We adopt an efficient semi-Lagrangian time integration scheme that is more accurate and computationally less expensive than the three-stage TVD Runge–Kutta integration. We apply our spatially fifth- and seventh-order schemes to a suite of simulations of collisionless self-gravitating systems and electrostatic plasma simulations, including linear and nonlinear Landau damping in one dimension and Vlasov–Poisson simulations in a six-dimensional phase space. The high-order schemes achieve a significantly improved accuracy in comparison with the third-order positive-flux-conserved scheme adopted in our previous study. With the semi-Lagrangian time integration, the computational cost of our high-order schemes does not significantly increase, but remains roughly the same as that of the third-order scheme. Vlasov–Poisson simulations on mesh grids have been successfully performed on a massively parallel computer.

First Demonstration of Electrostatic Damping of Parametric Instability at Advanced LIGO.

Interferometric gravitational wave detectors operate with high optical power in their arms in order to achieve high shot-noise limited strain sensitivity. A significant limitation to increasing the optical power is the phenomenon of three-mode parametric instabilities, in which the laser field in the arm cavities is scattered into higher-order optical modes by acoustic modes of the cavity mirrors. The optical modes can further drive the acoustic modes via radiation pressure, potentially producing an exponential buildup. One proposed technique to stabilize parametric instability is active damping of acoustic modes. We report here the first demonstration of damping a parametrically unstable mode using active feedback forces on the cavity mirror. A 15 538Hz mode that grew exponentially with a time constant of 182sec was damped using electrostatic actuation, with a resulting decay time constant of 23sec. An average control force of 0.03nN was required to maintain the acoustic mode at its minimum amplitude.

Improving the treatment of coarse-grain electrostatics: CVCEL.

We propose an analytic approach for calculating the electrostatic energy of proteins or protein complexes in aqueous solution. This method, termed CVCEL (Circular Variance Continuum ELectrostatics), is fitted to Poisson calculations and is able to reproduce the corresponding energies for different choices of solute dielectric constant. CVCEL thus treats both solute charge interactions and charge self-energies, and it can also deal with salt solutions. Electrostatic damping notably depends on the degree of solvent exposure of the charges, quantified here in terms of circular variance, a measure that reflects the vectorial distribution of the neighbors around a given center. CVCEL energies can be calculated rapidly and have simple analytical derivatives. This approach avoids the need for calculating effective atomic volumes or Born radii. After describing how the method was developed, we present test results for coarse-grain proteins of different shapes and sizes, using different internal dielectric constants and different salt concentrations and also compare the results with those from simple distance-dependent models. We also show that the CVCEL approach can be used successfully to calculate the changes in electrostatic energy associated with changes in protein conformation or with protein-protein binding.

Classical Description of the Vibrational Spectroscopy, Structure, and Electrostatics of the Halide Solvation Shell with the POLIR Potential.

POLIR (POLarizable for IR) (J. Chem. Phys. 2008, 129, 034504) is a polarizable, flexible, and transferable water potential which describes the IR spectrum in the O-H stretch region via the classical dipole correlation function with simple quantum corrections. POLIR also reproduces experimental spectral shifts in solutions of Ca(2+), Mg(2+), and Cu(2+) (J. Am. Chem. Soc. 2011, 133, 9441-9450). Here we present an extended investigation of POLIR water in the solvation shell of the halides F(-), Cl(-), Br(-), and I(-) using various interaction potentials, polarizabilities, and short-range electrostatic damping parameters. Our results indicate that the correlation of the first solvation shell dipoles produces IR spectra that are in agreement with experiment; that is, vibrational spectral shifts may be obtained with classical mechanics and simple corrections. Calculated ion induced dipoles agree with quantum simulations. Further analysis shows that ion-dependent shifts in the spectra may be attributed to the hydrogen bond O···O-H angle distribution within the first solvation shell, decomposed into ion-water and water-water contributions.

Stability and Dynamic Analysis of the Electrostatic MEMS Actuators

The nonlinear dynamic behavior of the micro actuators is analyzed in this paper. In general, analysis of the electrostatic device is quite difficult and complicated due to the electrostatic coupling effect and the nonlinear electrostatic force. In this study, a hybrid method for the micro-structure system, which combines the differential transformation and finite difference approximation techniques, is used to overcome the nonlinear electrostatic coupling phenomenon. The analysis takes account of the electrostatic coupling effect, the fringing field effect, the residual stress, the nonlinear electrostatic force and squeeze-film damping effect. Finally, the actuating conditions which ensure the stability of the micro actuators are identified by reference to phase portraits.

Dynamic Analysis of Micro Devices with Squeeze-Film Damping Effect Using Hybrid Numerical Scheme

Using traditional methods such as perturbation theory or Galerkin approach method to analyze the dynamic response of electrostatic devices is not easy due to the complexity of the interactions between the electrostatic coupling effect, the fringing field effect, the residual stress, the nonlinear electrostatic force and squeeze-film damping effect. Accordingly, the present study proposes a new approach for analyzing the dynamic response of such devices using a hybrid numerical scheme comprising the differential transformation method and the finite difference method by pure DC or combined DC / AC loading. The validity of the proposed scheme is confirmed by comparing the results obtained for the pull-in voltage of the micro-beam with those presented in the literature derived using a variety of schemes. Overall, the results show that the hybrid numerical scheme provides a suitable means of analyzing the nonlinear dynamic behavior of a wide variety of common electrostatically-actuated microstructures.

Interfacial behavior of simple inorganic salts at the air-water interface investigated with a polarizable model with electrostatic damping

New molecular models that incorporated polarizable interactions with electrostatic damping were developed to better understand the interfacial properties of aqueous electrolyte systems. The models were parameterized to give free energies of aqueous solvation and the change in activity with respect to concentration in agreement with experiment. Specifically, we investigated NaCl, NaBr, and NaI systems, finding anion propensity for the air-water interface was reduced in comparison with previously developed polarizable models. This coincided with a more negative surface excess than that given by previously developed polarizable models. Furthermore, we investigated the interfacial properties of SrCl2 aqueous systems, finding that strontium had a moderate enhancement in interfacial density in comparison with bulk, while still having a fairly large negative surface excess, in agreement with experimental results.

Pairwise Alternatives to Ewald Summation for Calculating Long-Range Electrostatics in Ionic Liquids.

Room temperature ionic liquid calculations require extensive sampling due to the large degree of localized structuring in the liquid phase relative to conventional solutions. Consequently, a large amount of computer time is required for the convergence of solvent properties, much of which is spent evaluating long-range electrostatics via Ewald summations. The damped Coulomb potential and cutoff-neutralized method of Wolf et al. (J. Chem. Phys.1999, 110, 8254) provides the framework for an accurate, linear-scaling alternative to Ewald in the ionic liquid simulations. The method has been the subject of multiple modifications for improved accuracy, including the damped Coulombic potential of Zahn et al. (J. Phys. Chem. B2002, 106, 10725), the damped shifted force method of Fennell and Gezelter (J. Chem. Phys.2006, 124, 234104), and the shifted force gradient of Kale and Herzfeld (J. Chem. Theory Comput.2011, 7, 3620). These pairwise electrostatic interaction alternatives along with the CHARMM shifted force potential and a new method proposed herein, the shifted force third derivative (SF3), have been examined on 59 unique ionic liquid combinations of 1-alkyl-3-methylimidazolium [RMIM] (R = M (methyl), E (ethyl), B (butyl), H (hexyl), and O (octyl)) and N-alkylpyridinium [RPyr] cations, along with Cl(-), PF6(-), BF4(-), NO3(-), AlCl4(-), Al2Cl7(-), and TfO(-) anions. Monte Carlo simulations utilizing our custom OPLS-AA ionic liquid force field and employing the pairwise alternatives with multiple cutoff distances and electrostatic damping values are compared to the energetics from full Ewald sums.

Dynamics of nonlinearly damped microcantilevers under electrostatic excitation

Nonlinear dynamic behaviour of a cantilever microbeam actuated by a combination of DC and AC loading are investigated in presence of squeeze-film damping. A reduced order model formulated accounting for the nonlinearities of the system arising out of electrostatic forces and squeeze-film damping is numerically simulated to observe the large amplitude dynamic characteristics near primary and superharmonic resonances. The emphasis is on the significance of nonlinear damping in capturing the true dynamic characteristics of microsystems formulated as distributed parameter model. The damping nonlinearity is found to considerably affect the dynamics with a profound stabilising effect on the microsystem. Under the effect of large DC bias voltage, frequency–response curves obtained for different amplitudes of AC excitation exhibit local and global bifurcations. Response sensitivity to initial conditions is investigated near bifurcation points. Findings in the superharmonic resonance domain are emphasised.

A High-Resolution Accelerometer With Electrostatic Damping and Improved Supply Sensitivity

In this paper, a charge-balancing accelerometer is presented. A hybrid interface topology is utilised to achieve high resolution, high linearity and low power supply sensitivity. The accelerometer consists of a micromechanical sensor element, a self-balancing bridge (SBB) open-loop readout, AC force feedback and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\Delta \Sigma$</tex> </formula> ADC. The SBB converts acceleration to ratiometric voltage. The ratiometric output of the SBB is converted to the digital domain by the ADC. In order to achieve high resolution, a micromechanical sensor element with a high quality factor, Q, is utilised. The AC force feedback is used for damping the high Q to get a low settling time. The sensor interface is fabricated in a standard 0.35 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\mu$</tex> </formula> m CMOS process. The fabricated chip has an area of 6.66 mm <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$^2$</tex></formula> and consumes 1 mA at a nominal supply voltage of 3.6 V. The sensor has a maximum DC nonlinearity of 1.3% over the commercial temperature range with an input range of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\pm} $</tex></formula> 1.15 g. The noise floor of the sensor is around 2 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\mu{\rm g}/\sqrt{{\rm Hz}}$</tex></formula> and the signal bandwidth is 200 Hz. The bias instability is 13 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\mu$</tex> </formula> g and the sensor gain variation is less than 5% in the 3–3.6 V supply range.